Construction 3.31 Fix \(n \ge 1\) . For any strictly totally disconnected perfectoid space \(Y\) , the usual derived category \(D(Y_{\mathop {\mathrm {\acute{e}t}}},\mathbf {Z}/\ell ^n)\)  is left-complete and identifies with a full subcategory \(D(Y_v,\mathbf {Z}/\ell ^n)\)  via pullback along \(Y_v \rightarrow Y_{\mathop {\mathrm {\acute{e}t}}}\) . Moreover, containment in this subcategory can be checked \(v\) -locally (see [1]}).
